Since this is partly a heritage trip, I won't tell you not to go to Sicily for just five days. I do suggest, though, that you slow down and concentrate your time in places where there's a family connection. Lucky for you, those are great spots!
Palermo is a hectic city with wild traffic, but it has wonderful sights, primarily in the form of lovely churches and oratorios. The cathedral at Monreale (just a bit out of town) is spectacular.
Does your husband have a family connection to Marettimo Island? I haven't seen it, but I made quick visits to the other two Egadi Islands and found them underwhelming for someone not interested in lying on a sandy beach for hours. This would not be my choice of destination on such a brief trip to Sicily, absent a family connection. Marettimo is not a short ferry ride even from Trapani. I'm pretty sure it's not possible as a day-trip from Palermo due to the infrequency of the ferries. I'd encourage you to spend a couple of nights in Trapani to make that trip easier (if it's essential) and allow time both to visit the very pretty historic district in Trapani itself and to see the very evocative and old hill-town of Erice that's just outside Trapani. I think Erice is one of Sicily's top sights.
I would not devote a day to the Mt. Etna excursion from Palermo myself, unless there's a family connection there. Better to have more time for the northwest corner of the island than to spread yourself so thin and spend a great deal of time on the Sicilian highways. In the unlikely event that you have extra time, there are other sightseeing possibilities much closer to Palermo.
I hope you'll have another opportunity to visit Sicily and see more of the island. It's worth 2 or 3 weeks all by itself.
After Sicily, I'd travel to Naples then Rome. I think Pompeii and the Amalfi area will require separate days, and I'm not sure that will leave much time for Naples and family visits, so you might want three days for the stay in Naples.
I don't really think you have time for both Bologna and Venice after Rome, so I'd drop Venice on this trip. Bologna itself is very nice, plus there are many lovely smaller towns that can be easily visited from Bologna on day trips.